Newform orbit 243.4.a.f

• Label: 243.4.a.f
• Level: $243$
• Weight: $4$
• Character orbit: 243.a
• Self dual: yes
• Analytic conductor: $14.337$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.3374641314\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{15}) \)
Defining polynomial:	\( x^{2} - 15 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^2 + 7 * q^4 + 2*b * q^5 + 23 * q^7 - b * q^8 + 30 * q^10 + 10*b * q^11 - 31 * q^13 + 23*b * q^14 - 71 * q^16 - 12*b * q^17 + 107 * q^19 + 14*b * q^20 + 150 * q^22 + 56*b * q^23 - 65 * q^25 - 31*b * q^26 + 161 * q^28 - 38*b * q^29 + 221 * q^31 - 63*b * q^32 - 180 * q^34 + 46*b * q^35 - 37 * q^37 + 107*b * q^38 - 30 * q^40 - 22*b * q^41 + 131 * q^43 + 70*b * q^44 + 840 * q^46 - 158*b * q^47 + 186 * q^49 - 65*b * q^50 - 217 * q^52 - 114*b * q^53 + 300 * q^55 - 23*b * q^56 - 570 * q^58 - 214*b * q^59 + 242 * q^61 + 221*b * q^62 - 377 * q^64 - 62*b * q^65 + 440 * q^67 - 84*b * q^68 + 690 * q^70 - 78*b * q^71 - 790 * q^73 - 37*b * q^74 + 749 * q^76 + 230*b * q^77 - 1057 * q^79 - 142*b * q^80 - 330 * q^82 - 104*b * q^83 - 360 * q^85 + 131*b * q^86 - 150 * q^88 - 90*b * q^89 - 713 * q^91 + 392*b * q^92 - 2370 * q^94 + 214*b * q^95 + 1193 * q^97 + 186*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 14 * q^4 + 46 * q^7 + 60 * q^10 - 62 * q^13 - 142 * q^16 + 214 * q^19 + 300 * q^22 - 130 * q^25 + 322 * q^28 + 442 * q^31 - 360 * q^34 - 74 * q^37 - 60 * q^40 + 262 * q^43 + 1680 * q^46 + 372 * q^49 - 434 * q^52 + 600 * q^55 - 1140 * q^58 + 484 * q^61 - 754 * q^64 + 880 * q^67 + 1380 * q^70 - 1580 * q^73 + 1498 * q^76 - 2114 * q^79 - 660 * q^82 - 720 * q^85 - 300 * q^88 - 1426 * q^91 - 4740 * q^94 + 2386 * q^97

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−3.87298
3.87298

−3.87298

0

7.00000

−7.74597

0

23.0000

3.87298

0

30.0000

1.2

3.87298

0

7.00000

7.74597

0

23.0000

−3.87298

0

30.0000

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( +1 \)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.4.a.f	✓	2
3.b	odd	2	1	inner	243.4.a.f	✓	2
9.c	even	3	2		243.4.c.f		4
9.d	odd	6	2		243.4.c.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.4.a.f	✓	2	1.a	even	1	1	trivial
243.4.a.f	✓	2	3.b	odd	2	1	inner
243.4.c.f		4	9.c	even	3	2
243.4.c.f		4	9.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(243))\):

\( T_{2}^{2} - 15 \)

\( T_{7} - 23 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 15 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 60 \)
$7$	\( (T - 23)^{2} \)
$11$	\( T^{2} - 1500 \)